The Tree Formula for MHV Graviton Amplitudes

TL;DR

This paper introduces a new tree-based formula for the tree-level n-graviton MHV amplitude that is genuinely gravitational in spirit, lacking cyclic ordering and exhibiting manifest ${S}_{n-2}$ symmetry with term-by-term $1/z^2$ large-$z$ behavior. The MHV tree formula expresses the amplitude as a sum over labeled tree diagrams with edge factors $[a\,b]/\langle a\,b\rangle$ and vertex factors $(\langle a\,n-1\rangle\langle a\,n\rangle)^{\deg(a)-2}$, multiplied by an overall $1/\langle n-1\,n\rangle^2$; an equivalent form absorbs vertex factors into propagators. The authors also develop a twistor-space link representation that clarifies cancellations and structure, and provide a proof using a BCFW-like deformation and induction on $n$, showing that the deformed amplitude has the correct residues and large-$z$ behavior. The work connects to prior formulas (e.g., Mason–Skinner and Bern–1998sv), highlights novel gravitational structure, and opens questions about fully $S_n$-symmetric formulations, extensions to non-MHV, and possible twistor-string interpretations with implications for loop amplitudes in supergravity.

Abstract

We present and prove a formula for the MHV scattering amplitude of n gravitons at tree level. Some of the more interesting features of the formula, which set it apart as being significantly different from many more familiar formulas, include the absence of any vestigial reference to a cyclic ordering of the gravitons--making it in a sense a truly gravitational formula, rather than a recycled Yang-Mills result, and the fact that it simultaneously manifests both S_{n-2} symmetry as well as large-z behavior that is O(1/z^2) term-by-term, without relying on delicate cancellations. The formula is seemingly related to others by an enormous simplification provided by O(n^n) iterated Schouten identities, but our proof relies on a complex analysis argument rather than such a brute force manipulation. We find that the formula has a very simple link representation in twistor space, where cancellations that are non-obvious in physical space become manifest.

Figures (1)

• Figure 1: All factorizations contributing to the on-shell recursion relation for the $n$-point MHV amplitude. Only the first diagram contributes to the residue at $z = \langle 1\,3\rangle/\langle 2\,3\rangle$.